published as:
Observation of the Dalitz decay η′→γe^{+}e^{-}
M. Ablikim et al. (BESIII Collaboration)
Phys. Rev. D 92, 012001 — Published 1 July 2015
DOI:
10.1103/PhysRevD.92.012001
M. Ablikim1, M. N. Achasov9,a, X. C. Ai1, O. Albayrak5, M. Albrecht4, D. J. Ambrose44, A. Amoroso48A,48C, F. F. An1,
Q. An45, J. Z. Bai1, R. Baldini Ferroli20A, Y. Ban31, D. W. Bennett19, J. V. Bennett5, M. Bertani20A, D. Bettoni21A,
J. M. Bian43, F. Bianchi48A,48C, E. Boger23,h, O. Bondarenko25, I. Boyko23, R. A. Briere5, H. Cai50, X. Cai1, O. Cakir40A,b,
A. Calcaterra20A, G. F. Cao1, S. A. Cetin40B, J. F. Chang1, G. Chelkov23,c, G. Chen1, H. S. Chen1, H. Y. Chen2, J. C. Chen1, M. L. Chen1, S. J. Chen29, X. Chen1, X. R. Chen26, Y. B. Chen1, H. P. Cheng17, X. K. Chu31, G. Cibinetto21A,
D. Cronin-Hennessy43, H. L. Dai1, J. P. Dai34, A. Dbeyssi14, D. Dedovich23, Z. Y. Deng1, A. Denig22, I. Denysenko23, M. Destefanis48A,48C, F. De Mori48A,48C, Y. Ding27, C. Dong30, J. Dong1, L. Y. Dong1, M. Y. Dong1, S. X. Du52, P. F. Duan1, J. Z. Fan39, J. Fang1, S. S. Fang1, X. Fang45, Y. Fang1, L. Fava48B,48C, F. Feldbauer22, G. Felici20A,
C. Q. Feng45, E. Fioravanti21A, M. Fritsch14,22, C. D. Fu1, Q. Gao1, X. Y. Gao2, Y. Gao39, Z. Gao45, I. Garzia21A,
C. Geng45, K. Goetzen10, W. X. Gong1, W. Gradl22, M. Greco48A,48C, M. H. Gu1, Y. T. Gu12, Y. H. Guan1, A. Q. Guo1, L. B. Guo28, Y. Guo1, Y. P. Guo22, Z. Haddadi25, A. Hafner22, S. Han50, Y. L. Han1, X. Q. Hao15, F. A. Harris42, K. L. He1,
Z. Y. He30, T. Held4, Y. K. Heng1, Z. L. Hou1, C. Hu28, H. M. Hu1, J. F. Hu48A,48C, T. Hu1, Y. Hu1, G. M. Huang6,
G. S. Huang45, H. P. Huang50, J. S. Huang15, X. T. Huang33, Y. Huang29, T. Hussain47, Q. Ji1, Q. P. Ji30, X. B. Ji1, X. L. Ji1, L. L. Jiang1, L. W. Jiang50, X. S. Jiang1, J. B. Jiao33, Z. Jiao17, D. P. Jin1, S. Jin1, T. Johansson49, A. Julin43,
N. Kalantar-Nayestanaki25, X. L. Kang1, X. S. Kang30, M. Kavatsyuk25, B. C. Ke5, R. Kliemt14, B. Kloss22,
O. B. Kolcu40B,d, B. Kopf4, M. Kornicer42, W. K¨uhn24, A. Kupsc49, W. Lai1, J. S. Lange24, M. Lara19, P. Larin14, C. Leng48C, C. H. Li1, Cheng Li45, D. M. Li52, F. Li1, G. Li1, H. B. Li1, J. C. Li1, Jin Li32, K. Li13, K. Li33, Lei Li3, P. R. Li41, T. Li33, W. D. Li1, W. G. Li1, X. L. Li33, X. M. Li12, X. N. Li1, X. Q. Li30, Z. B. Li38, H. Liang45, Y. F. Liang36,
Y. T. Liang24, G. R. Liao11, D. X. Lin14, B. J. Liu1, C. X. Liu1, F. H. Liu35, Fang Liu1, Feng Liu6, H. B. Liu12, H. H. Liu1,
H. H. Liu16, H. M. Liu1, J. Liu1, J. P. Liu50, J. Y. Liu1, K. Liu39, K. Y. Liu27, L. D. Liu31, P. L. Liu1, Q. Liu41, S. B. Liu45, X. Liu26, X. X. Liu41, Y. B. Liu30, Z. A. Liu1, Zhiqiang Liu1, Zhiqing Liu22, H. Loehner25, X. C. Lou1,e, H. J. Lu17,
J. G. Lu1, R. Q. Lu18, Y. Lu1, Y. P. Lu1, C. L. Luo28, M. X. Luo51, T. Luo42, X. L. Luo1, M. Lv1, X. R. Lyu41, F. C. Ma27,
H. L. Ma1, L. L. Ma33, Q. M. Ma1, S. Ma1, T. Ma1, X. N. Ma30, X. Y. Ma1, F. E. Maas14, M. Maggiora48A,48C, Q. A. Malik47, Y. J. Mao31, Z. P. Mao1, S. Marcello48A,48C, J. G. Messchendorp25, J. Min1, T. J. Min1, R. E. Mitchell19,
X. H. Mo1, Y. J. Mo6, C. Morales Morales14, K. Moriya19, N. Yu. Muchnoi9,a, H. Muramatsu43, Y. Nefedov23, F. Nerling14,
I. B. Nikolaev9,a, Z. Ning1, S. Nisar8, S. L. Niu1, X. Y. Niu1, S. L. Olsen32, Q. Ouyang1, S. Pacetti20B, P. Patteri20A, M. Pelizaeus4, H. P. Peng45, K. Peters10, J. Pettersson49, J. L. Ping28, R. G. Ping1, R. Poling43, Y. N. Pu18, M. Qi29,
S. Qian1, C. F. Qiao41, L. Q. Qin33, N. Qin50, X. S. Qin1, Y. Qin31, Z. H. Qin1, J. F. Qiu1, K. H. Rashid47, C. F. Redmer22,
H. L. Ren18, M. Ripka22, G. Rong1, X. D. Ruan12, V. Santoro21A, A. Sarantsev23,f, M. Savri´e21B, K. Schoenning49, S. Schumann22, W. Shan31, M. Shao45, C. P. Shen2, P. X. Shen30, X. Y. Shen1, H. Y. Sheng1, W. M. Song1, X. Y. Song1,
S. Sosio48A,48C, S. Spataro48A,48C, G. X. Sun1, J. F. Sun15, S. S. Sun1, Y. J. Sun45, Y. Z. Sun1, Z. J. Sun1, Z. T. Sun19,
C. J. Tang36, X. Tang1, I. Tapan40C, E. H. Thorndike44, M. Tiemens25, D. Toth43, M. Ullrich24, I. Uman40B, G. S. Varner42,
B. Wang30, B. L. Wang41, D. Wang31, D. Y. Wang31, K. Wang1, L. L. Wang1, L. S. Wang1, M. Wang33, P. Wang1, P. L. Wang1, Q. J. Wang1, S. G. Wang31, W. Wang1, X. F. Wang39, Y. D. Wang20A, Y. F. Wang1, Y. Q. Wang22, Z. Wang1,
Z. G. Wang1, Z. H. Wang45, Z. Y. Wang1, T. Weber22, D. H. Wei11, J. B. Wei31, P. Weidenkaff22, S. P. Wen1, U. Wiedner4,
M. Wolke49, L. H. Wu1, Z. Wu1, L. G. Xia39, Y. Xia18, D. Xiao1, Z. J. Xiao28, Y. G. Xie1, Q. L. Xiu1, G. F. Xu1, L. Xu1, Q. J. Xu13, Q. N. Xu41, X. P. Xu37, L. Yan45, W. B. Yan45, W. C. Yan45, Y. H. Yan18, H. X. Yang1, L. Yang50, Y. Yang6,
Y. X. Yang11, H. Ye1, M. Ye1, M. H. Ye7, J. H. Yin1, B. X. Yu1, C. X. Yu30, H. W. Yu31, J. S. Yu26, C. Z. Yuan1,
W. L. Yuan29, Y. Yuan1, A. Yuncu40B,g, A. A. Zafar47, A. Zallo20A, Y. Zeng18, B. X. Zhang1, B. Y. Zhang1, C. Zhang29, C. C. Zhang1, D. H. Zhang1, H. H. Zhang38, H. Y. Zhang1, J. J. Zhang1, J. L. Zhang1, J. Q. Zhang1, J. W. Zhang1,
J. Y. Zhang1, J. Z. Zhang1, K. Zhang1, L. Zhang1, S. H. Zhang1, X. Y. Zhang33, Y. Zhang1, Y. H. Zhang1, Y. T. Zhang45,
Z. H. Zhang6, Z. P. Zhang45, Z. Y. Zhang50, G. Zhao1, J. W. Zhao1, J. Y. Zhao1, J. Z. Zhao1, Lei Zhao45, Ling Zhao1,
M. G. Zhao30, Q. Zhao1, Q. W. Zhao1, S. J. Zhao52, T. C. Zhao1, Y. B. Zhao1, Z. G. Zhao45, A. Zhemchugov23,h, B. Zheng46, J. P. Zheng1, W. J. Zheng33, Y. H. Zheng41, B. Zhong28, L. Zhou1, Li Zhou30, X. Zhou50, X. K. Zhou45, X. R. Zhou45,
X. Y. Zhou1, K. Zhu1, K. J. Zhu1, S. Zhu1, X. L. Zhu39, Y. C. Zhu45, Y. S. Zhu1, Z. A. Zhu1, J. Zhuang1, L. Zotti48A,48C, B. S. Zou1, J. H. Zou1
(BESIII Collaboration)
1 Institute of High Energy Physics, Beijing 100049, People’s Republic of China 2 Beihang University, Beijing 100191, People’s Republic of China
3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s Republic of China 4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA 6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 China Center of Advanced Science and Technology, Beijing 100190, People’s Republic of China
8 COMSATS Institute of Information Technology, Lahore, Defence Road, Off Raiwind Road, 54000 Lahore, Pakistan 9 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090, Russia
10GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany 11 Guangxi Normal University, Guilin 541004, People’s Republic of China
12 GuangXi University, Nanning 530004, People’s Republic of China 13 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
14 Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz, Germany 15 Henan Normal University, Xinxiang 453007, People’s Republic of China
16 Henan University of Science and Technology, Luoyang 471003, People’s Republic of China 17Huangshan College, Huangshan 245000, People’s Republic of China
18Hunan University, Changsha 410082, People’s Republic of China 19 Indiana University, Bloomington, Indiana 47405, USA
20(A)INFN Laboratori Nazionali di Frascati, I-00044, Frascati, Italy; (B)INFN and University of Perugia, I-06100, Perugia,
Italy
21 (A)INFN Sezione di Ferrara, I-44122, Ferrara, Italy; (B)University of Ferrara, I-44122, Ferrara, Italy 22Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz, Germany
23 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
24 Justus Liebig University Giessen, II. Physikalisches Institut, Heinrich-Buff-Ring 16, D-35392 Giessen, Germany 25 KVI-CART, University of Groningen, NL-9747 AA Groningen, The Netherlands
26Lanzhou University, Lanzhou 730000, People’s Republic of China 27Liaoning University, Shenyang 110036, People’s Republic of China 28 Nanjing Normal University, Nanjing 210023, People’s Republic of China
29 Nanjing University, Nanjing 210093, People’s Republic of China 30Nankai University, Tianjin 300071, People’s Republic of China
31 Peking University, Beijing 100871, People’s Republic of China 32Seoul National University, Seoul, 151-747 Korea 33Shandong University, Jinan 250100, People’s Republic of China 34Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
35 Shanxi University, Taiyuan 030006, People’s Republic of China 36 Sichuan University, Chengdu 610064, People’s Republic of China
37 Soochow University, Suzhou 215006, People’s Republic of China 38Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
39Tsinghua University, Beijing 100084, People’s Republic of China
40 (A)Istanbul Aydin University, 34295 Sefakoy, Istanbul, Turkey; (B)Dogus University, 34722 Istanbul, Turkey; (C)Uludag
University, 16059 Bursa, Turkey
41 University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China 42 University of Hawaii, Honolulu, Hawaii 96822, USA
43 University of Minnesota, Minneapolis, Minnesota 55455, USA 44University of Rochester, Rochester, New York 14627, USA
45 University of Science and Technology of China, Hefei 230026, People’s Republic of China 46 University of South China, Hengyang 421001, People’s Republic of China
47 University of the Punjab, Lahore-54590, Pakistan
48 (A)University of Turin, I-10125, Turin, Italy; (B)University of Eastern Piedmont, I-15121, Alessandria, Italy; (C)INFN,
I-10125, Turin, Italy
49 Uppsala University, Box 516, SE-75120 Uppsala, Sweden 50Wuhan University, Wuhan 430072, People’s Republic of China 51Zhejiang University, Hangzhou 310027, People’s Republic of China 52Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at the Novosibirsk State University, Novosibirsk, 630090, Russia bAlso at Ankara University, 06100 Tandogan, Ankara, Turkey
c Also at the Moscow Institute of Physics and Technology, Moscow 141700, Russia and at the Functional Electronics
Laboratory, Tomsk State University, Tomsk, 634050, Russia
dCurrently at Istanbul Arel University, 34295 Istanbul, Turkey e Also at University of Texas at Dallas, Richardson, Texas 75083, USA f Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
g Also at Bogazici University, 34342 Istanbul, Turkey
hAlso at the Moscow Institute of Physics and Technology, Moscow 141700, Russia
We report the first observation of the Dalitz decay η′
→ γe+e−
, based on a data sample of 1.31 billion J/ψ events collected with the BESIII detector. The η′
mesons are produced via the J/ψ → γη′
decay process. The ratio Γ(η′
→ γe+e−
)/Γ(η′
→ γγ) is measured to be (2.13 ± 0.09(stat.) ± 0.07(sys.)) × 10−2. This corresponds to a branching fraction B(η′
→ γe+e−
) = (4.69 ± 0.20(stat.) ± 0.23(sys.)) × 10−4. The transition form factor is extracted and different expressions
are compared to the measured dependence on the e+e−
invariant mass. The results are consistent with the prediction of the Vector Meson Dominance model.
I. INTRODUCTION
Electromagnetic (EM) Dalitz decays of light
pseu-doscalar mesons, P → γl+l−
(P = π0, η, η′
; l = e, µ), play an important role in revealing the structure of hadrons and the interaction mechanism between photons and hadrons [1]. If one assumes point-like particles, the decay rates can be exactly calculated by Quantum Elec-trodynamics (QED) [2]. Modifications to the QED decay rate due to the inner structure of the mesons are encoded
in the transition form factor (TFF) F (q2), where q is the
momentum transferred to the lepton pair, and q2 is the
square of the invariant mass of the lepton pair. A recent summary and discussion of this subject can be found in Ref. [3].
The knowledge of the TFF is also important in
stud-ies of the muon anomalous magnetic moment, aµ =
(gµ−2)/2, which is the most precise low-energy test of the
Standard Model (SM) and an important probe for new physics [4, 5]. The theoretical uncertainty on the SM
calculation of aµ is dominated by hadronic corrections
and therefore limited by the accuracy of their determi-nation. In particular, the hadronic light-by-light (HLbL)
scattering contribution to aµincludes two
meson-photon-photon vertices that can be related to the form factors in
P → γγ∗
→ γe+e−
decays [5]. Thus, models describing these transitions should be tested as precisely as possi-ble to reduce the uncertainty in the SM prediction for
(gµ− 2)/2.
In this work, the Dalitz decay η′
→ γe+e−
is measured for the first time. The differential decay width,
normal-ized to the radiative decay width η′
→ γγ, is [1] dΓ(η′ → γl+l− ) dq2Γ(η′ → γγ) = 2α 3π 1 q2 s 1 −4m 2 l q2 1 + 2m 2 l q2 1 − q 2 m2 η′ !3 F (q2) 2 = [QED(q2 )] × |F (q2 )|2, (1)
where mη′ and ml are the masses of the η′ meson and
the lepton, respectively; α is the fine structure constant;
and [QED(q2)] represents the calculable QED part for a
point-like meson. The TFF, F (q2), which is described by
phenomenological models, can be experimentally deter-mined from differences between the measured di-lepton invariant mass spectrum and the QED calculation. In the Vector Meson Dominance (VMD) model [6], it is assumed that interactions between virtual photon and hadrons are dominated by a superposition of neutral vector meson states. One commonly used expression for the multi-pole form factor is [7]: F (q2) = NX V gη′γV 2gV γ · m2 V m2 V − q2− iΓVmV , (2)
where N is a normalization constant ensuring that
F (0) = 1; V = ρ, ω, φ; mV, ΓV are the masses and
widths of these vector mesons; and gη′γV and gV γ are
the corresponding coupling constants.
The parameter to be experimentally determined is the slope of the form factor b, which is related to the effective virtual vector meson mass Λ by
b = dF dq2 q2=0= Λ −2 (3) In experiments, the single-pole form factor is generally used to extract the slope of the form factor. For the case
of the η′
, the pole is expected to lie within the kinematic boundaries of the decay. The square of the form factor is described by
|F (q2
)|2= Λ2(Λ2+ γ2)
(Λ2− q2)2+ Λ2γ2 (4)
where the parameters Λ and γ correspond to the mass and width of the Breit-Wigner shape for the effective contributing vector meson. To a first approximation, Λ ≈
Mρ≈ 0.7 GeV and γ ≈ Γρ≈ 0.12 GeV.
For the η′
Dalitz decay, only the process η′
→ γµ+µ−
has been observed and the slope of the form factor was
measured to be bη′ = (1.7 ± 0.4) GeV−2 [1, 8]. To date,
the process η′
→ γe+e−
has not been observed yet. The most stringent upper limit on the ratio of decay widths
Γ(η′
→ γe+e−
)/Γ(η′
→ γγ) is 4.1 × 10−2 at the 90%
confidence level (CL) from the CLEO Collaboration [9],
which is above the predicted value of (2.06 ± 0.02) × 10−2
from the modified VMD model [10].
In the VMD model, the TFF slope is expected to
be bη′ = 1.45 GeV
−2 [11, 12], while for chiral
pertur-bation theory it is bη′ = 1.60 GeV
−2 [13]. A recent
calculation based on a dispersion integral gives bη′ =
1.53+0.15−0.08 GeV
−2 [14].
We report the first observation of the η′
→ γe+e−
decay and the extraction of the TFF. The source of the
η′
mesons are radiative J/ψ → γη′
decays in a sample
of 1.31 billion J/ψ events (2.25 × 108 events were taken
in 2009 [15] and 1.09 × 109 in 2012) [16] collected by
the BESIII [17] at the BEPCII e+e−
collider. The η′
→ γγ decay events in the same data sample are used for normalization.
II. THE BESIII EXPERIMENT AND MONTE CARLO SIMULATION
BEPCII is a double-ring multi-bunch e+e−
collider running in the tau-charm energy region. The BESIII detector, described in detail in Ref. [17], has a geomet-rical acceptance of 93% of 4π solid angle. It consists of a drift chamber (MDC), a time-of-flight (TOF) system, and an electromagnetic calorimeter (EMC), all enclosed in a super-conducting solenoid with 1.0 T (0.9 T in 2012) magnetic field. The small-cell helium based MDC pro-vides the tracking of charged particle and ionization en-ergy loss (dE/dx) measurement. The single cell position
resolution is 130 µm and the transverse momentum reso-lution is 0.5% at 1 GeV/c. The TOF system for particle identification (PID) is made of plastic scintillators. It has 80 ps time resolution in the barrel, and 110 ps in the end caps. The EMC is made of 6240 CsI (Tl) crys-tals. The energy resolution is 2.5% in the barrel and 5% in the end caps for 1.0 GeV photons. Outside the
solenoid, a muon chamber system made of 1272 m2
resis-tive plate chambers detects muon tracks with momenta greater than 0.5 GeV/c.
The geant4-based [18] simulation software BOOST includes the description of geometry and material of the BESIII detector, the detector response and digitization models, and also tracks the detector running conditions and performance. A Monte Carlo (MC) simulated sample of 1.2 billion J/ψ inclusive decays is used to study poten-tial backgrounds. The production of the J/ψ resonance is simulated by the MC event generator kkmc [19]; the known decay modes are generated by evtgen [20, 21] with branching fractions set at the world average val-ues [22], while unknown decays are generated by
lund-charm [23]. The evtgen package is used to generate
J/ψ → γη′
, η′
→ γe+e−
and η′
→ γγ events. The decay
J/ψ → γη′
is generated with an angular distribution of
1 + cos2θ
γ, where θγ is the radiative photon angle
rela-tive to the positron beam direction in the J/ψ rest frame.
In generating η′
→ γe+e−
, the TFF is parameterized by the multi-pole VMD model in Eq.(2) with the parameters taken from Ref. [1].
III. SIGNAL SELECTION: J/ψ → γη′
, η′
→ γe+e−
Charged tracks are reconstructed from hits registered in the MDC. Only tracks with | cos θ| < 0.93 are re-tained, where θ is the polar angle with respect to the
beam axis. The tracks are required to pass within
10 cm of the center of the interaction region in the beam direction (Z axis) and within 1 cm in the plane perpendicular to the beam. Event candidates are re-quired to have two well reconstructed charged tracks with net charge zero. For electron identification, information from dE/dx and TOF is combined to compute probabil-ities for the electron (Prob(e)) and pion (Prob(π)) hy-pothesis. To separate electrons from pions, we require Prob(e)/(Prob(e) + Prob(π)) > 0.95. Final states with kaons cannot contribute to the background because of the limited phase space.
Electromagnetic showers are reconstructed from clus-ters of energy deposits in the EMC. The photon candi-date showers must have a minimum energy of 25 MeV in the barrel region (| cos θ| < 0.80) and 50 MeV in the end cap region (0.86 < | cos θ| < 0.92). Showers in the region between the barrel and the end caps are poorly measured and excluded from the analysis. To exclude charged-particle induced activities, the showers are re-quired to be separated from the extrapolated positions
of any charged track by at least 10◦
. In addition, cluster
timing requirements are used to suppress electronic noise and unrelated energy deposits.
In the analysis, all selection criteria are studied and optimized by MC simulations and QED control samples selected from data. A vertex fit is performed on the
elec-tron and posielec-tron tracks, and a loose χ2 requirement is
applied to ensure that they come from a common vertex. To improve resolution and reduce background, a
four-constraint (4C) kinematic fit is performed to the γγe+e−
hypothesis that constrains the total four-momentum of the detected particles to be equal to the initial four-momentum of the colliding beams. For events with more than two photon candidates, the combination with the
smallest χ2
4C is selected. Only events with χ
2 4C<100 are retained. For the J/ψ → γη′ , η′ → γe+e− signal channel, the largest background comes from QED processes and
J/ψ → e+e−
γγ decays. For these channels, the
com-bination of the e+e−
with any final-state photon
pro-duces a smooth M (γe+e−
) distribution. The QED
back-ground mainly comes from e+e−
→ e+e−
γγ and e+e−
→
3γ events in which one γ converts into an e+e−
pair.
These are studied using a e+e−
collision data sample of
2.92 fb−1 taken at√s = 3.773 GeV [24], which is
dom-inated by QED processes. For those processes, most of the photons have low energy and are at small angles rela-tive to the incoming electron or positron beam directions. To reduce this background, the energy of the low-energy photon is further required to be higher than 200 MeV, and the angle between the photon and the electron or positron initial direction in the final states is required to
be larger than 10◦
.
The primary peaking background comes from the
de-cay J/ψ → γη′
, η′
→ γγ followed by a γ conversion in the material in front of the MDC, including the beam pipe and the inner wall of the MDC. The distance from the re-constructed vertex point of the electron-positron pair to
the z axis, defined as δxy =
q
R2
x+ R2y, is used to
distin-guish γ conversion events from signal events [25], where
Rx and Ry are the distances in the x and y directions,
respectively. A scatter plot of Ry versus Rx is shown
in Fig. 1 (a) for MC-simulated J/ψ → γη′
, η′
→ γγ decays, in which one of the photons undergoes
conver-sion to an e+e−
pair. As indicated in Fig. 1 (a), the inner circle matches the position of the beam pipe, while the outer circle corresponds to the position of the inner
wall of the MDC. Figure 1 (b) shows the δxy
distribu-tions for the MC simulated J/ψ → γη′
, η′
→ γe+e−
,
J/ψ → γη′
, η′
→ γγ events, together with the selected
data events and events from the η′
mass sideband. The two peaks above 2.0 cm correspond to the photon
con-version of the γ from J/ψ → γη′
, η′
→ γγ events,
while the events near δxy = 0 cm originate from the
interaction point. We require δxy < 2 cm to suppress
the photon-conversion background, which retains about 80% of the signal events while the remaining photon-conversion events is about 5% the size of the signal. After
all selections, the normalized number of expected peaking
background events from J/ψ → γη′
, η′
→ γγ is 42.7±8.0, where the error is dominantly from the difference in se-lection efficiencies for the γ conversion events between data and MC.
Another possible source of peaking background is
J/ψ → γη′
, η′
→ γπ+π−
, where the two pions are
misidentified as an e+e−
pair. An exclusive MC sam-ple that includes coherent contributions from ρ, ω, and
the box anomaly in the decay η′
→ γπ+π−
[26] is used to study this background. We find that the kinematic fit to the electron-positron hypothesis shifts the spectrum
away from the η′
mass and, thus, the resulting M(γe+e−
)
distribution does not peak at the η′
mass value. The nor-malized number of events from this background source af-ter all selections is 9.7 ±0.4, which is negligible compared
to the non-peaking background from e+e−
→ e+e− γγ. Rx (cm) -10 -5 0 5 10 Ry (cm) -10 -5 0 5 10 beam pipe inner MDC wall
(a)
(cm) xy δ 0 2 4 6 8 10 Events/ (0.2 cm) 0 100 200 300 data signal MC Conv MC γ sideband MC+sideband(b)
Figure 1. Electron-positron vertex position distribution: (a) scatter plot of Ry versus Rx for MC-simulated J/ψ → γη′,
η′
→ γγ events. (b) δxy distributions. The (black) crosses
are data. The (red) dashed line shows the MC-simulated J/ψ → γη′
, η′
→ γe+e−
signal events. The (orange) dotted-dashed histogram shows the background from γ-conversion events. The (green) shaded area is estimated from the η′
mass sideband. The (blue) line is the sum of MC and the sideband estimate. In (b), the solid arrow indicates the requirement on δxy.
The combination of γe+e−
with invariant mass
clos-est to mη′ is taken to reconstruct the η
′
. The
result-ing M (γe+e−
) distribution after the selection criteria
is shown in Fig. 2 and exhibits a clear peak at the η′
mass. An unbinned extended maximum likelihood (ML) fit is performed to determine the signal yield. The signal probability density function (PDF) is represented by the signal MC shape. The shape for the non-peaking back-ground is described by a first-order Chebychev polyno-mial. The background yield and its PDF parameters are allowed to vary in the fit. The peaking background from
the γ-conversion of J/ψ → γη′
, η′
→ γγ decay is obtained from the MC-simulated shape with the yield fixed as
de-scribed before. The fitting range is 0.85−1.05 GeV/c2.
The net signal yield and the detection efficiency are sum-marized in Table I.
Table I. Number of observed signal events, Nη′→γe+e−
(Nη′→γγ), and detection efficiency, ǫη′→γe+e− (ǫη′→γγ) for
J/ψ → γη′ , η′ → γe+e− (J/ψ → γη′ , η′ → γγ). The un-certainties are statistical only.
η′ → γe+e− η′ → γγ Nη′→γe+e− (Nη′→γγ) 864 ± 36 70846 ± 292 ǫη′→γe+e− (ǫη′→γγ) 24.5% 42.8%
)
2) (GeV/c
-e
+e
γ
M(
0.85 0.9 0.95 1 1.05)
2Events/ (5 MeV/c
0 100 200 300)
2) (GeV/c
-e
+e
γ
M(
0.85 0.9 0.95 1 1.05)
2Events/ (5 MeV/c
0 100 200 300Figure 2. Invariant γe+e−
mass distribution for the se-lected signal events. The (black) crosses are the data, the (red) dashed line represents the signal, the (green) dot-dashed curve shows the non-peaking background shapes, the (orange) shaded component is the shape of the J/ψ → γη′
, η′
→ γγ peaking background events. The total fit result is shown as the (blue) solid line.
IV. NORMALIZATION CHANNEL: J/ψ → γη′, η′→γγ
The decay J/ψ → γη′
, η′
→ γγ is studied using the same data set, and we quote our result in terms of the
ratio Γ(η′
→ γe+e−
)/Γ(η′
→ γγ). In this ratio the un-certainties due to the total number of J/ψ events and
the branching fraction for J/ψ → γη′
cancel, and the uncertainty due to the photon detection efficiency par-tially cancels.
Events with zero charged particles and at least three photon candidates are selected with the same require-ments that are used for the signal events. A 4C kine-matic fit is performed to the J/ψ → γγγ hypothesis. For events with more than three photons, the combination
with the smallest χ2
4C is selected. The χ
2
4C is required
to be less than 100. The two photon combination with
invariant mass M (γγ) closest to mη′ is taken as from the
η′
decay.
Detailed MC studies indicate that no peaking back-ground remains after all the selection criteria. The non-peaking background mainly comes from the continuum
process e+e−
→ γγγ and J/ψ → γπ0π0 decays. The
f0(1500), f0(1710), f0(2020), f2(1270), f4(2050). Because
the η′
decays isotropically, the angular distribution of
photons from the η′
decays is flat in cos θdecay, where
θdecay is the angle of the decay photon in the η′ helic-ity frame. In contrast, background events from QED
continuum processes and J/ψ → γπ0π0 decays tend to
accumulate near cos θdecay= ±1. We suppress these
non-peaking backgrounds by requiring | cos θdecay| < 0.8.
The M (γγ) distribution for events that survive the se-lection requirements is shown in Fig. 3. An unbinned ML
fit is performed to obtain the yield of J/ψ → γη′
, η′
→ γγ. The PDF used to represent the signal is taken from the MC, and the PDF for the non-peaking background is a first-order Chebychev polynomial with coefficients determined from the fit. The resulting signal yield and the MC-determined detection efficiency are summarized in Table I.
)
2) (GeV/c
γ
γ
M(
0.85 0.9 0.95 1 1.05)
2Events/ (5 MeV/c
0 5000 10000 15000)
2) (GeV/c
γ
γ
M(
0.85 0.9 0.95 1 1.05)
2Events/ (5 MeV/c
0 5000 10000 15000Figure 3. Invariant mass distribution of M (γγ) for the se-lected events in the normalization channel. The (black) crosses are data, the (red) dashed curves represents the η′
→ γγ signal, the (green) dot-dashed curve shows the non-peaking background. The fit result is shown as the (blue) solid curve.
V. SYSTEMATIC UNCERTAINTIES IN THE RELATIVE DECAY WIDTH
Table II lists all sources of systematic uncertainties
associated with the measurement of the ratio Γ(η′
→
γe+e−
)/Γ(η′
→ γγ). Most systematic uncertainties are determined from comparisons of low-background, high-statistics data samples with results from MC simulations. The electron and positron tracking and PID efficien-cies are determined using a sample of radiative Bhabha
e+e−
→ γe+e−
(including J/ψ → γe+e−
) events col-lected at the J/ψ energy. Differences in tracking and PID efficiencies between data and MC simulation are de-termined for every bin of a two-dimensional distribution of the momentum versus polar angle of the lepton tracks. These are used to determine an overall weighted differ-ence per track of (1.1 ± 0.3)% for the tracking efficiency and (1.9±0.3)% for the PID efficiency. The MC efficiency is corrected for these differences, and the uncertainties
Table II. Summary of relative systematic uncertainties for the determination of the ratio Γ(η′→γe+e−)
Γ(η′→γγ) . The last row is the
uncertainty associated with the J/ψ → γη′
, η′
→ γγ normal-ization sample.
Sources Systematic uncertainties(in %)
MDC tracking 0.6
PID 0.6
Photon detection 1.0
Veto of gamma conversion 1.8
4C kinematic fit 1.0
Form factor uncertainty 1.6 Fit range & Bkg shape 0.9 Uncertainty of Nη′→γγ 1.2
Total 3.3
of the correction coefficients are assigned as the system-atic uncertainties associated with the lepton tracking and PID efficiencies.
The systematic uncertainty associated with the photon detection efficiency is studied using three different meth-ods, as described in Ref. [27]. The three methods provide consistent results for the photon efficiency uncertainty to be 1% per photon. Because the systematic uncertainty
from the radiative photon and one photon from the η′
cancel in the ratio, the total systematic uncertainty from photon detection is 1%.
In the analysis, the peaking background from J/ψ →
γη′
, η′
→ γγ γ-conversion events is suppressed by the
requirement δxy< 2 cm. To estimate the systematic
un-certainty associated with this requirement, we use a
sam-ple of J/ψ → π+π−
π0, π0
→ γe+e−
that includes both
π0 Dalitz decays and π0
→ γγ decays with one photon externally converted to an electron-positron pair. The data-MC difference of 1.8% for these events is considered as the systematic uncertainty for our γ-conversion veto
requirement on δxy.
A systematic uncertainty associated with the kine-matic fit will occur if the track-helix parameters for data and MC simulated events are not consistent. Follow-ing the procedure described in Ref. [28], we use the
J/ψ → π+π−
π0, π0
→ γe+e−
decay as a control sample to extract the correction factors from the pull distribu-tions of the track helix parameters. The 1% difference between the efficiencies with and without helix parame-ter corrections is taken as the systematic uncertainty.
To estimate the systematic uncertainty due to the effi-ciency dependence on the choice of form-factor param-eterizations, signal MC events are also generated us-ing a sus-ingle-pole VMD model, shown in Eq. (4), with Λ = (0.79 ± 0.05) GeV and γ = (0.13 ± 0.06) GeV,
which are taken from the fitted results described below in Section VII. The relative difference in the detection efficiency compared to that of the multi-pole model is taken as the uncertainty associated with the form-factor parameterization.
In the fit to the γe+e−
mass distribution, the signal PDF is fixed to the signal MC shape. An alternative fit is performed by using a convolution of a MC signal shape with a Gaussian function that is used to describe the MC-data difference due to the resolution. The fitted width of the Gaussian is (0.39 ± 0.19) MeV, and the fit yields 863.8 ± 36.0 signal events. The difference from the nominal fit is negligible. Finally, the uncertainty due to the non-peaking background shape is estimated by varying the PDF shape and fitting range in the ML fit. The changes in yields for these variations give systematic uncertainties due to these backgrounds.
The systematic uncertainty in the measurement of
J/ψ → γη′
, η′
→ γγ associated with the uncertainty from the kinematic fit is estimated using a control sample of
e+e−
→ γγγ at 3.650 GeV [29] and found to be less than 1%. The uncertainty for this channel due to back-ground is estimated to be less than 0.3% from variations in the PDF shape and fitting range. The uncertainty
from the requirement | cos θdecay| < 0.8 is 0.4%. When
combined with the 0.4% statistical uncertainty, the total
uncertainty associated with Nη′→γγ is 1.2%.
Assuming all systematic uncertainties in Table II are independent, the total systematic uncertainty, obtained from their quadratic sum, is 3.3%.
VI. RELATIVE DECAY WIDTH
The ratio Γ(η′
→ γe+e−
)/Γ(η′
→ γγ) is determined using the following formula:
Γ(η′ → γe+e− ) Γ(η′ → γγ) = Nη′→γe+e− Nη′→γγ · ǫη′→γγ ǫη′→γe+e− , (5)
where Nη′→γe+e− (Nη′→γγ) and ǫη′→γe+e− (ǫη′→γγ) are
the number of observed signal events and the
detec-tion efficiency, respectively, for J/ψ → γη′
, η′
→ γe+e−
(J/ψ → γη′
, η′
→ γγ) decays, as listed in Table I. The result is Γ(η′ → γe+e− ) Γ(η′ → γγ) = (2.13±0.09(stat.)±0.07(sys.))×10 −2. (6) Using the η′
→ γγ branching fraction value listed in
PDG [22], we obtain the first measurement of the η′
→ γe+e− branching fraction of B(η′ → γe+e− ) = (4.69 ± 0.20(stat.) ± 0.23(sys.)) × 10−4. (7)
VII. FORM FACTOR MEASUREMENT
The TFF is extracted from the bin-by-bin efficiency
corrected signal yields for eight different M (e+e−
) bins.
The bin widths are all chosen to be 0.1 GeV/c2. Since
this is much wider than the M (e+e−
) resolution, which
is 5∼6 MeV/c2 depending on M (e+e−
), no unfolding is
needed. The signal yield in each M (e+e−
) bin i is
ob-tained by performing bin-by-bin fits to the M (γe+e−
) mass distributions using the fitting procedure described in Section III. The peaking background from the J/ψ →
γη′
, η′
→ γγ only exists in the first bin, and the yield is fixed to the normalized number in the bin. The fit
re-sults are shown in Fig. 4. The fitted (nobs
i ) and
efficiency-corrected signal yields (ncorr
i ) for each Mi(e+e−) bin are
summarized in Table III. Figure 5 shows the
efficiency-corrected signal yields versus M (e+e−
) with the QED shape superimposed for comparison. The discrepancy between QED and data, which reflects the TFF, is
evi-dent in the high M (e+e−
) region.
The systematic uncertainties on ncorr
i include the
un-certainty from the MDC tracking efficiency, PID, photon detection, kinematic fit, veto of gamma conversion, back-ground description and signal shape; they are the same as those described in Section V.
The partial ratio ri = ∆Γ(η′ → γe+e−)i/Γ(η′ → γγ)
for each given M (e+e−
) bin i, is defined as ri≡ ∆Γ(η′ → γe+e− )i Γ(η′ → γγ) = ncorr i ǫη′→γγ Nη′→γγ , (8) where ∆Γ(η′ → γe+e−
)i is the integrated rate in each
M (e+e−
) interval.
The result for |F |2in each M (e+e−
) bin is obtained by
dividing the value ri by the integrated QED predication
in each M (e+e−
) interval (see Eq. (1)). The values of
|F |2for each M (e+e−
) bin are summarized in Table IV. A variety of models have been traditionally used to parameterize the TFF. The most common one, based on VMD [6], uses only the first term in the dispersion re-lation. In this single pole model, the TFF is given by Eq. (4). The results of a least-squares fit with the single pole model is shown in Fig. 6; the parameters of the form
factors are determined to be Λη′ = (0.79 ± 0.05) GeV,
γη′ = (0.13 ± 0.06) GeV. From the fitted value of the
parameter Λη′, the slope of the form factor is obtained
to be (1.60 ± 0.19) GeV−2, in agreement with the
re-sult bη′ = (1.7 ± 0.4) GeV
−2 obtained in the process of
η′
→ γµ+µ−
[1].
To test the robustness of the slope extracted from the
simple pole model, we also fit the data below 0.5 GeV/c2
using the single pole Ansatz used in lighter meson studies:
F (q2) = 1
(1 − q2/Λ2), (9)
The parameterization diverges at M (e+e−
) = Λ and, therefore, can not be used for the whole kinematic region. The result of this fit is shown in Fig. 7. The slope of the
) 2 ) (GeV/c -e + e γ M( 0.85 0.9 0.95 1 1.05 ) 2 Events/ (5 MeV/c 0 50 100 150 200 ) 2 ) (GeV/c -e + e γ M( 0.85 0.9 0.95 1 1.05 ) 2 Events/ (5 MeV/c 0 50 100 150 200 ) -e + M(e 2 0~100 MeV/c ) 2 ) (GeV/c -e + e γ M( 0.85 0.9 0.95 1 1.05 ) 2 Events/ (8 MeV/c 0 10 20 30 40 50 ) 2 ) (GeV/c -e + e γ M( 0.85 0.9 0.95 1 1.05 ) 2 Events/ (8 MeV/c 0 10 20 30 40 50 ) -e + M(e 2 100~200 MeV/c ) 2 ) (GeV/c -e + e γ M( 0.85 0.9 0.95 1 1.05 ) 2 Events/ (8 MeV/c 0 10 20 30 40 ) 2 ) (GeV/c -e + e γ M( 0.85 0.9 0.95 1 1.05 ) 2 Events/ (8 MeV/c 0 10 20 30 40 ) -e + M(e 2 200~300 MeV/c ) 2 ) (GeV/c -e + e γ M( 0.85 0.9 0.95 1 1.05 ) 2 Events/ (12 MeV/c 0 20 40 ) 2 ) (GeV/c -e + e γ M( 0.85 0.9 0.95 1 1.05 ) 2 Events/ (12 MeV/c 0 20 40 ) -e + M(e 2 300~400 MeV/c ) 2 ) (GeV/c -e + e γ M( 0.85 0.9 0.95 1 1.05 ) 2 Events/ (12 MeV/c 0 20 40 ) 2 ) (GeV/c -e + e γ M( 0.85 0.9 0.95 1 1.05 ) 2 Events/ (12 MeV/c 0 20 40 ) -e + M(e 2 400~500 MeV/c ) 2 ) (GeV/c -e + e γ M( 0.85 0.9 0.95 1 1.05 ) 2 Events/ (12 MeV/c 0 10 20 30 40 ) 2 ) (GeV/c -e + e γ M( 0.85 0.9 0.95 1 1.05 ) 2 Events/ (12 MeV/c 0 10 20 30 40 ) -e + M(e 2 500~600 MeV/c ) 2 ) (GeV/c -e + e γ M( 0.85 0.9 0.95 1 1.05 ) 2 Events/ (12 MeV/c 0 10 20 30 40 ) 2 ) (GeV/c -e + e γ M( 0.85 0.9 0.95 1 1.05 ) 2 Events/ (12 MeV/c 0 10 20 30 40 ) -e + M(e 2 600~700 MeV/c ) 2 ) (GeV/c -e + e γ M( 0.85 0.9 0.95 1 1.05 ) 2 Events/ (12 MeV/c 0 10 20 30 40 ) 2 ) (GeV/c -e + e γ M( 0.85 0.9 0.95 1 1.05 ) 2 Events/ (12 MeV/c 0 10 20 30 40 ) -e + M(e 2 700~800 MeV/c
Figure 4. Results from bin-by-bin fits to the M (γe+e−
) distributions for different M (e+e−
) bins. The (black) crosses are data, the (red) dashed curves represent the signal, the (green) dot-dashed curves show the non-peaking backgrounds, the (orange) shaded component for the M (e+e−
) < 100 MeV/c2bin is the shape of the peaking background from J/ψ → γη′
, η′
→ γγ. The total fit results are shown as (blue) solid curves.
Table III. Fitted (nobs
i ) and efficiency-corrected (ncorri ) signal yields for the eight M (e+e −
) bins, and ratios (ri). The
uncer-tainties are statistical only.
M (e+e− ) (GeV/c2) [0.0, 0.1] [0.1, 0.2] [0.2, 0.3] [0.3, 0.4] nobs i 545 ± 27 86.5 ± 10.7 62.1 ± 9.8 45.6 ± 9.7 ncorri 2380 ± 120 368 ± 46 194 ± 31 128 ± 27 ri (10 −2 ) 1.44 ± 0.07 0.22 ± 0.03 0.12 ± 0.02 0.08 ± 0.02 M (e+e− ) (GeV/c2) [0.4, 0.5] [0.5, 0.6] [0.6, 0.7] [0.7, 0.8] nobs i 45.4 ± 9.0 29.9 ± 8.0 28.0 ± 7.8 25.0 ± 6.9 ncorr i 135 ± 27 93.3 ± 25.0 96.2 ± 26.8 109 ± 30 ri (10 −2) 0.08 ± 0.02 0.06 ± 0.02 0.06 ± 0.02 0.07 ± 0.02
Table IV. Values of |F |2 in each M (e+e−
) bin, where the first uncertainties are statistical and the second ones systematic.
M(e+e− ) (GeV/c2) [0.0, 0.1] [0.1, 0.2] [0.2, 0.3] [0.3, 0.4] |F |2 1.05 ± 0.05 ± 0.03 1.12 ± 0.14 ± 0.04 1.16 ± 0.18 ± 0.05 1.33 ± 0.28 ± 0.05 M(e+e− ) (GeV/c2) [0.4, 0.5] [0.5, 0.6] [0.6, 0.7] [0.7, 0.8] |F |2 2.48 ± 0.49 ± 0.25 3.30 ± 0.88 ± 0.31 7.66 ± 2.13 ± 0.89 26.6 ± 7.3 ± 1.9
)
2) (GeV/c
-e
+M(e
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8)
2Events/ (0.1 GeV/c
1 10 2 10 3 10 Experiment QED calculationFigure 5. Efficiency-corrected signal yields ncorr i versus
M (e+e−
). The (black) crosses are data and the (gray) shaded histogram indicates the point-like QED result.
form factor is determined to be bη′ = (1.58±0.34) GeV
−2, which is in good agreement with the result of (1.60 ±
0.19) GeV−2 using Eq. (4).
The quadratic difference between the uncertainties of the parameters with only statistical errors used in the fits and the uncertainties of the parameters with com-bined statistical and systematic errors used in the fits is taken as the systematic uncertainty on the param-eters. The resulting parameters in Eq. (4) are
deter-mined to be Λη′ = (0.79 ± 0.04(stat.) ± 0.02(sys.)) GeV,
γη′ = (0.13 ± 0.06(stat.) ± 0.03(sys.)) GeV, respectively.
VIII. SUMMARY
In summary, with a sample of 1.31 billion J/ψ events collected in the BESIII detector, we have made the
first measurement of the EM Dalitz decay process η′
→
γe+e−
and measure the ratio Γ(η′
→ γe+e−
)/Γ(η′
→
γγ) = (2.13 ± 0.09(stat.) ± 0.07(sys.)) × 10−2.
Us-ing the PDG value for the η′
→ γγ branching
frac-tion [22], we determine B(η′
→ γe+e−
) = (4.69 ±
0.20(stat.) ± 0.23(sys.)) × 10−4. We present
measure-ments of the TFF as a function of M (e+e−
). Our
TFF results can be described with a single pole pa-rameterization Eq. (4), with mass and width
parame-ters of Λη′ = (0.79 ± 0.04(stat.) ± 0.02(sys.)) GeV, and
γη′ = (0.13 ± 0.06(stat.) ± 0.03(sys.)) GeV, respectively.
The slope of the TFF corresponds to (1.60 ±0.17(stat.)±
0.08(sys.)) GeV−2 and agrees within errors with the
VMD model predictions. The uncertainty of the η′
tran-sition form factor slope matches the best determination in the space-like region from the CELLO collaboration
bη′ = (1.60 ± 0.16) GeV−2 [30], and improves the
pre-vious determination of the slope in the time-like region
bη′ = (1.7 ± 0.4) GeV
−2 [1, 8]. The η′
form factor is
determined by both universal π+π−
rescattering and a reaction specific part, with the latter contributing about 20% to the form factor slope [14]. Therefore our result
is sensitive specifically to the η′
internal EM structure.
In addition, the decay η′
→ γe+e−
is closely related to
η′
→ γπ+π−
, and in particular the transition form factor could be predicted from the invariant mass distribution of
the two pions and the branching ratio of the η′
→ γπ+π−
decay in a model independent way using a dispersive inte-gral. Also, the knowledge of the TFF is useful for studies of the HLbL scattering contribution to the muon
anoma-lous magnetic moment, aµ= (gµ− 2)/2 [5].
IX. ACKNOWLEDGEMENT
The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong
sup-port. This work is supported in part by National
Key Basic Research Program of China under
Con-tract No. 2015CB856700; National Natural Science
Foundation of China (NSFC) under Contracts Nos. 11125525, 11235011, 11322544, 11335008, 11425524; the Chinese Academy of Sciences (CAS) Large-Scale Scien-tific Facility Program; Joint Large-Scale ScienScien-tific Facil-ity Funds of the NSFC and CAS under Contracts Nos. 11179007, U1232201, U1332201; CAS under Contracts Nos. KJCX2-YW-N29, KJCX2-YW-N45; 100 Talents Program of CAS; INPAC and Shanghai Key Labora-tory for Particle Physics and Cosmology; German
Re-search Foundation DFG under Contract No.
Collab-orative Research Center CRC-1044; Istituto Nazionale di Fisica Nucleare, Italy; Ministry of Development of Turkey under Contract No. DPT2006K-120470; Russian Foundation for Basic Research under Contract No.
14-)
2) (GeV/c
-e
+M(e
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 2)|
2|F(q
1 10Figure 6. Fit to the single pole form factor |F |2using Eq. (4).
The (black) crosses are data, where the statistical and system-atic uncertainties are combined, the (blue) solid curve shows the fit results. The (gray) dotted line shows to the point-like case (i.e. with |F |2= 1) for comparison.
)
2) (GeV/c
-e
+M(e
0 0.1 0.2 0.3 0.4 0.5 2)|
2|F(q
0.5 1 1.5 2 2.5 3 3.5Figure 7. Determination of the form factor slope by fitting to |F |2 using the single pole form factor of Eq. (9). The (black) crosses are data, where the uncertainties are the combined statistical and systematic uncertainties, the (blue) solid curve shows the fit result. The (gray) dotted line corresponds to the point-like particle case (with |F |2= 1).
07-91152; U. S. Department of Energy under Contracts Nos. FG02-04ER41291, FG02-05ER41374, DE-FG02-94ER40823, DESC0010118; U.S. National Sci-ence Foundation; University of Groningen (RuG) and the Helmholtzzentrum fuer Schwerionenforschung GmbH (GSI), Darmstadt; WCU Program of National Research Foundation of Korea under Contract No. R32-2008-000-10155-0
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